Stochastic Boundary Research Articles

Probabilistic Models and Statistics for Electronic Financial Markets in the Digital Age

The scope of this manuscript is to review some recent developments in statistics for discretely observed semimartingales which are motivated by applications for financial markets. Our journey through this area stops to take closer looks at a few selected topics discussing recent literature. We moreover highlight and explain the important role played by some classical concepts of probability and statistics. We focus on three main aspects: Testing for jumps; rough fractional stochastic volatility; and limit order microstructure noise. We review jump tests based on extreme value theory and complement the literature proposing new statistical methods. They are based on asymptotic theory of order statistics and the Rényi representation. The second stage of our journey visits a recent strand of research showing that volatility is rough. We further investigate this and establish a minimax lower bound exploring frontiers to what extent the regularity of latent volatility can be recovered in a more general framework. Finally, we discuss a stochastic boundary model with one-sided microstructure noise for high-frequency limit order prices and its probabilistic and statistical foundation.

Image driven deep learning method with FFT solver for predicting the microscale full field stress of stochastic boundary composites

AbstractA novel image‐driven deep learning approach embedded with model‐data‐knowledge information can directly predict the full field stress distribution and mechanical properties of composites solely based on initial scanning geometry images. The fiber spatial distribution and morphological features are identified by Scanning Electron Microscopy (SEM) initially. An effective random fiber generation algorithm is further utilized to generate images database comprising equivalent geometric models of various microstructures. Subsequently, the geometry model images database is analyzed by fast Fourier transform (FFT) to produce the database including the elastic modulus parameters and full field stress distributions of composites with distinct microstructures. Finally, an innovative image‐learning comprehensive paradigm uniting convolutional neural networks and convolutional autoencoders is elaborated systematically to learn the inherent laws of geometry images and field images. The results show that the proposed method have capacity to effectively and accurately predict the elastic modulus and full field stress distributions combined with error analysis even for stochastic boundary composites. The proposed methodology is a symbiosis of cutting‐edge image learning and non‐destructive testing techniques for composites, which can directly use the local non‐destructive or scanning images of composite structural components to quickly obtain field information and further predict damage evolution online.Highlights Proposing a novel deep learning approach combined with FFT directly to predict stress distribution of stochastic boundary composites solely based on micro‐images. Adopting equivalent geometric models dispersed by higher pixels mesh density considering extreme thin interface. Incorporate adaptive algorithms for streamlined training optimization. An innovative integration of deep learning and non‐destructive testing technology for the stress distribution and properties prediction online.

Theory guided Lagrange programming neural network for subsurface flow problems

A deep learning model can perform efficient uncertainty quantification (UQ) for reservoir flow with uncertain model parameters, but usually requires large amounts of training data to ensure accuracy. However, the cost of obtaining large amounts of data is prohibitive, and the performance will deteriorate if sufficient training data is lacking. Alternatively, more interpretable neural networks with embedded physical laws have recently been used to solve partial differential equations as well as to solve UQ problems. This approach has received a lot of attention due to its low data volume requirements and its adherence to the laws of physics during the training process. In this paper, we propose a theory-guided framework based on a bilevel programming model with hard constraints to embed physical meaning in the model. Theory guided Lagrange programming neural network (TGLPNN) combines the method of Lagrange programming neural network approach where physical laws such as stochastic partial differential equations and boundary conditions are incorporated into the training process of a convolutional neural network. At the same time, the upper-level variables are iteratively optimized. The method based on Lagrange programming neural network inherently embeds physical laws in the network. Practical applications have shown that TGLPNN can provide higher prediction accuracy compared to state-of-the-art physics-driven methods and improved efficiency compared to numerical methods.

Space‐time stochastic Galerkin boundary elements for acoustic scattering problems

SummaryAcoustic emission or scattering problems naturally involve uncertainties about the sound sources or boundary conditions. This article initiates the study of time domain boundary elements for such stochastic boundary problems for the acoustic wave equation. We present a space‐time stochastic Galerkin boundary element method which is applied to sound‐hard, sound‐soft and absorbing scatterers. Uncertainties in both the sources and the boundary conditions are considered using a polynomial chaos expansion. The numerical experiments illustrate the performance and convergence of the proposed method in model problems and present an application to a problem from traffic noise.

A Boundary Control Problem for Stochastic 2D-Navier–Stokes Equations

AbstractWe study a stochastic velocity tracking problem for the 2D-Navier–Stokes equations perturbed by a multiplicative Gaussian noise. From a physical point of view, the control acts through a boundary injection/suction device with uncertainty, modeled by stochastic non-homogeneous Navier-slip boundary conditions. We show the existence and uniqueness of the solution to the state equation, and prove the existence of an optimal solution to the control problem.

Global well-posedness and interior regularity of 2D Navier–Stokes equations with stochastic boundary conditions

The paper is devoted to the analysis of the global well-posedness and the interior regularity of the 2D Navier–Stokes equations with inhomogeneous stochastic boundary conditions. The noise, white in time and coloured in space, can be interpreted as the physical law describing the driving mechanism on the atmosphere–ocean interface, i.e. as a balance of the shear stress of the ocean and the horizontal wind force.

Combination of Karhunen-Loève and intrusive polynomial chaos for uncertainty quantification of thermomagnetic convection problem with stochastic boundary condition

Uncertainty propagation analysis plays a crucial role in understanding the impact of variations in initial condition, boundary condition, and fluid physical properties on simulation results for thermomagnetic convection heat transfer problems. To effectively analyze the uncertainty problem of thermomagnetic convection caused by random temperature fluctuations, a novel and comprehensive computational framework based on intrusive polynomial chaos approach was proposed in this paper. Our proposed approach aims to represent random temperature boundaries using Karhunen-Loève expansion (KLE) and stochastic output response using polynomial chaos expansion (PCE). In addition, we use spectral projection method to transform the stochastic control equation into a group of deterministic control equations. We obtained the uncertainty characteristics of the stochastic output response by solving each polynomial chaos basis. Compared with Monte Carlo simulation (MCS), our approach accurately and efficiently predicts the uncertainty characteristics of thermomagnetic convection problems under stochastic temperature boundary conditions while significantly reducing computational resources.

Basin delimitation and stability assessment for power systems in the projective space

This paper presents a method for determining the global basin boundary of the power systems. To show the general outline of the basin, the power system is shrink to the projective space. First, a new vector field is induced according to Poincare’s method. After the transformation, the structural stability of the system remains the same. The singularities at infinity are obtained by the proposed shrinking-projection method. The boundary of the basin which is extended to the infinity are complemented by the singularities at infinity in the projective space. The stability of the power system is then evaluated based on the distribution of the singularities at infinity. Finally, the proposed method is applied to the power system with perturbations to study the characteristic of the stochastic basin boundary. The validity of the proposed method is demonstrated on a 9-bus, 11-bus and 68-bus test system.

Reduced-Order Modeling with Time-Dependent Bases for PDEs with Stochastic Boundary Conditions

Reduced-Order Modeling with Time-Dependent Bases for PDEs with Stochastic Boundary Conditions

Research on second level of vulnerability criteria of parametric rolling with stochastic stability theory

The second level of vulnerability criteria of parametric rolling proposed by the International Maritime Organization (IMO) is aimed to evaluate the probability of roll instability and large roll motion of ships in the designated sea areas. Compared with the numerical simulation in random waves, the stability boundary derived from the criteria has large errors. Some additional checks are required in the key region. A new method is proposed to assess ship parametric roll in the designed sea area based on stochastic stability theory. A nonlinear system has the same stability properties as its equivalent linear system. Therefore, the nonlinear parametric rolling motion equation is linearized in order to study its stochastic stability in random waves. Based on the top Lyapunov exponent and the stochastic averaging theory, the expressions for parametric rolling stability boundary are given. The accuracy of these stability boundaries is verified by comparison with numerical simulation. The new method with the stochastic stability boundary can quickly and accurately evaluate the probability of large roll motion of ships in the designated sea areas which illustrates the possibility of improvement to a certain extent.

Stochastic Vibrations of a System of Plates Immersed in Fluid Using a Coupled Boundary Element, Finite Element, and Finite Difference Methods Approach.

The main objective of this work is to investigate the natural vibrations of a system of two thin (Kirchhoff-Love) plates surrounded by liquid in terms of the coupled Stochastic Boundary Element Method (SBEM), Stochastic Finite Element Method (SFEM), and Stochastic Finite Difference Method (SFDM) implemented using three different probabilistic approaches. The BEM, FEM, and FDM were used equally to describe plate deformation, and the BEM was applied to describe the dynamic interaction of water on a plate surface. The plate's inertial forces were expressed using a diagonal or consistent mass matrix. The inertial forces of water were described using the mass matrix, which was fully populated and derived using the theory of double-layer potential. The main aspect of this work is the simultaneous application of the BEM, FEM, and FDM to describe and model the problem of natural vibrations in a coupled problem in solid-liquid mechanics. The second very important novelty of this work is the application of the Bhattacharyya relative entropy apparatus to test the safety of such a system in terms of potential resonance. The presented concept is part of a solution to engineering problems in the field of structure and fluid dynamics and can also be successfully applied to the analysis of the dynamics of the control surfaces of ships or aircraft.

Exponential mixing of Vlasov equations under the effect of gravity and boundary

In this paper, we study exponentially fast mixing induced/enhanced by gravity and stochastic boundary in the kinetic theory of Vlasov equations. We consider the Vlasov equations with and without a vertical magnetic field inside a horizontally-periodic 3D half-space equipped with a non-isothermal diffusive reflection boundary condition of bounded continuous boundary temperature at the bottom. We construct both stationary solutions and global-in-time dynamical solutions in L∞. We prove that moments of a dynamical fluctuation around the steady solutions decay exponentially fast in L∞. As a key of this proof, we establish a uniform bound of so-called residual measures independently of the bouncing number of stochastic characteristics, by constructing a continuous stationary outgoing boundary flux which is strictly positive almost everywhere.

Mechanism of interdigitation formation at apical boundary of MDCK cell

It has been reported that the MDCK cell tight junction shows stochastic fluctuation and forms the interdigitation structure, but the mechanism of the pattern formation remains to be elucidated. In the present study, we first quantified the shape of the cell-cell boundary at the initial phase of pattern formation. We found that the Fourier transform of the boundary shape shows linearity in the log-log plot, indicating the existence of scaling. Next, we tested several working hypotheses and found that the Edwards-Wilkinson equation, which consists of stochastic movement and boundary shortening, can reproduce the scaling property. Next, we examined the molecular nature of stochastic movement and found that myosin light chain puncta may be responsible. Quantification of boundary shortening indicates that mechanical property change may also play some role. Physiological meaning and scaling properties of the cell-cell boundary are discussed.

Risk assessment of a layered slope considering spatial variabilities of interlayer and intralayer

While many studies have analyzed the effects of spatial variabilities of intralayer soil parameters on the probability of failure and risk of layered slopes, there is a paucity of research on the effects of interlayer variability. This paper is intended to investigate the influence of interlayer variability on the mechanical responses of a layered slope, as well as the effects of spatial variabilities of strength parameters of intralayer soils. To characterize the uncertainty of intralayer and interlayer spatial variabilities of layered geologic profiles, an integrated random field is developed using the program of FLAC based on both the two-dimensional random field theory and one-dimensional random process theory. Particularly, the random field for the interlayer transition zone is generated by a linear interpolation technique. Within the framework of Monte-Carlo simulation, a large number of stochastic calculations for a two-layered slope are conducted to evaluate the influences of spatial variabilities of both interlayer and intralayer on the stability and risk of the slope. The results show the failure mode of a two-layered slope changes greatly due to the stochastic layered boundary. It is also noted that the probability and risk of slope failure might be underestimated if ignoring the uncertainty arising from layered boundary and layered transition zone. The proposed equations can provide helpful information when investigating the probability and risk associated with layered slopes.

Desenvolvimento regional da agricultura familiar: Cooperativismo e associativismo

We sought to assess the impact of cooperatives and associations on Brazilian family agricultural production.A stochastic spatial boundary was estimated. Then, variables that explained the technicalproductive efficiency were studied. The regional percentage of establishments linked to cooperativesand associations had a positive effect on local production and development. The Northeast showedlower technical efficiency than the South, Southeast and Midwest regions. In the South, the institutionalenvironment managed to promote policies with greater local integration (bottom-up) and, in theNortheast, it depended on more centralized actions with low insertion of local institutions (top-down).Schooling and technical assistance contributed to efficiency gains.

Побудова критерію довготривалого руйнування втоми для тонкостінних шаруватих оболонок

A model and criterion of long-term fatigue failure for thin-walled layered shells is built, taking into account the influence of the type of stress state. The problem of calculating the number of cycles to failure under combined loading is considered. Solutions are built on the basis of the concept of equivalent stresses. The problem of determining local stresses in composites of random structure is formulated within the framework of the second-order nonlinear theory. The solution of the stochastic boundary value problem on determining the stress concentration in a unidirectional composite with a metal matrix (MMC) was obtained. To build a complete system of equations of the second order, the method of successive approximations is used. The parameters of the stress concentration at the boundary of the components are determined. The given examples show the importance of the influence of nonlinear properties on the redistribution of stresses near the fibers. The possibility of predicting the long-term strength of the material is shown. The necessary information about the material for the formulation of failure criteria is the S-N curves for individual components of the combined stresses.

Inflation with stochastic boundary

We study the Brownian motion of a field where there are boundaries in the inflationary field space. Both the field and the boundary undergo Brownian motions with the amplitudes of the noises determined by the Hubble expansion rate of the corresponding de Sitter spacetime. This setup mimics models of inflation in which curvature perturbation is induced from inhomogeneities generated at the surface of the end of inflation. The cases of the drift-dominated regime as well as the diffusion-dominated regime are studied in detail. We calculate the first hitting probabilities as well as the mean number of $e$-folds for the field to hit either of the boundaries in the field space. The implications for models of inflation are reviewed.

On equilibrating non-periodic molecular dynamics samples for coupled particle-continuum simulations of amorphous polymers

In the context of fracture simulations of polymers, the molecular mechanisms in the vicinity of the crack tip are of particular interest. Nevertheless, to keep the computational cost to a minimum, a coarser resolution must be used in the remaining regions of the numerical sample. For the specific case of amorphous polymers, the Capriccio method bridges the gap between the length and time scales involved at the different levels of resolution by concurrently coupling molecular dynamics (MD) with the finite element method (FEM). Within the scope of the Capriccio approach, the coupling to the molecular MD region introduces non-periodic, so-called stochastic boundary conditions (SBC). In similarity to typical simulations under periodic boundary conditions (PBC), the SBC MD simulations must reach an equilibrium state before mechanical loads are exerted on the coupled systems. In this contribution, we hence extensively study the equilibration properties of non-periodic MD samples using the Capriccio method. We demonstrate that the relaxation behavior of an MD-FE coupled MD domain utilizing non-periodic boundary conditions is rather insensitive to the specific coupling parameters of the method chosen to implement the boundary conditions. The behavior of an exemplary system equilibrated with the parameter set considered as optimal is further studied under uniaxial tension and we observe some peculiarities in view of creep and relaxation phenomena. This raises important questions to be addressed in the further development of the Capriccio method.

Uncertainty quantification of two-phase flow in porous media via the Coupled-TgNN surrogate model

The uncertainty quantification (UQ) of subsurface two-phase flow usually requires numerous executions of forward simulations under varying conditions. In this work, a novel coupled theory-guided neural network (TgNN) based surrogate model is built to facilitate computation efficiency under the premise of satisfactory accuracy. The core notion of this proposed method is to bridge two separate blocks on top of an overall network. They underlie the TgNN model in a coupled form, which reflects the coupling nature of pressure and water saturation in the two-phase flow equation. The TgNN model not only relies on labeled data, but also incorporates underlying scientific theory and experiential rules (e.g., governing equations, stochastic parameter fields, boundary and initial conditions, well conditions, and expert knowledge) as additional components into the loss function. The performance of the TgNN-based surrogate model for two-phase flow problems is tested by different numbers of labeled data and collocation points, as well as the existence of data noise. The proposed TgNN-based surrogate model offers an effective way to solve the coupled nonlinear two-phase flow problem, and shows good accuracy and strong robustness when compared with the purely data-driven surrogate model. By combining the accurate TgNN-based surrogate model with the Monte Carlo method, UQ tasks can be performed at a minimum cost to evaluate statistical quantities. Since the heterogeneity of the random fields strongly impacts the results of the surrogate model, corresponding variance and correlation length are added to the input of the neural network to maintain its predictive capacity. In addition, several more complicated scenarios are also considered, including dynamically changing well conditions and dynamically changing variance of random fields. The results show that the TgNN-based surrogate model exhibits satisfactory accuracy, stability, and efficiency in the UQ problem of subsurface two-phase flow.

Stochastic Collocation Method for Stochastic Optimal Boundary Control of the Navier–Stokes Equations

Stochastic Collocation Method for Stochastic Optimal Boundary Control of the Navier–Stokes Equations